A Characterization of Valuation Domains via m-Canonical Ideals # Ayman Badawi * Department of Mathematics and Statistics, American University of Sharjah, Sharjah, United Arab Emirates ABSTRACT A nonzero ideal I of an intergral domain R is said to be an m-canonical ideal of R if ðI : ðI : J ÞÞ ¼ J for every nonzero ideal J of R. In this paper, we show that if a quasi-local integral domain ðR; MÞ admits a proper m-canonical ideal I of R, then the following statements are equivalent: (1) R is a valuation domain. (2) I is a divided m-canonical ideal of R. (3) cM ¼ I for some nonzero c 2 R. (4) ðI : MÞ is a principal ideal of R. (5) ðI : MÞ is an invertible ideal of R. (6) R is an integrally closed domain and ðI : MÞ is a finitely generated of R. (7) ðM : MÞ¼ R and ðI : MÞ is a finitely generated of R. (8) If J ¼ðI : MÞ, then J is a finitely generated of R and ðJ : J Þ¼ R. Among the many results in this paper, we show that an integral domain R is a valuation domain if and only if R admits a divided proper m-canonical ideal, # Communicated by I. Swanson. *Correspondence: Ayman Badawi, Department of Mathematics and Statistics, American University of Sharjah, P.O. Box 26666, Sharjah, United Arab Emirates; Fax: 00971-6-515- 2950; E-mail: abadawi@ aus.ac.ae. COMMUNICATIONS IN ALGEBRA Õ Vol. 32, No. 11, pp. 4363–4374, 2004 4363 DOI: 10.1081/AGB-200034159 0092-7872 (Print); 1532-4125 (Online) Copyright # 2004 by Marcel Dekker, Inc. www.dekker.com